A nonsmooth Morse-Sard theorem for subanalytic functions

According to the Morse-Sard theorem. any sufficiently smooth function on a Euclidean space remains constant along any arc of critical points. We prove here a theorem of Morse-Sard type suitable as a tool in variational analysis: we broaden the definition of a critical point to the standard notion in nonsmooth optimization. while we restrict the functions under consideration to be semialgebraic or subanalytic. We make no assumption of subdifferential regularity. Lojasiewiez-type inequalities for nonsmooth functions follow quickly from tools of the kind we develop, leading to convergence theory for subgradient dynamical systems. (c) 2005 Elsevier Inc. All rights reserved.